Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-24T03:20:19.676Z Has data issue: false hasContentIssue false
  • English
  • Français

The dead leaves model: a general tessellation modeling occlusion

Part of: General convexity Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Charles Bordenave ,
Yann Gousseau  and
François Roueff
Charles Bordenave*
Affiliation:
Ecole Normale Supérieure and INRIA
Yann Gousseau*
Affiliation:
Télécom Paris
François Roueff*
Affiliation:
Télécom Paris
*
∗ Postal address: Département d'Informatique, Ecole Normale Supérieure, 45 rue d'Ulm, F-75230 Paris Cedex 05, France. Email address: [email protected]
∗∗ Postal address: Département Traitement du Signal et des Images, CNRS UMR 5141, ENST, 46 rue Barrault, 75634 Paris Cedex 13, France.
∗∗ Postal address: Département Traitement du Signal et des Images, CNRS UMR 5141, ENST, 46 rue Barrault, 75634 Paris Cedex 13, France.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this article, we study a particular example of general random tessellation, the dead leaves model. This model, first studied by the mathematical morphology school, is defined as a sequential superimposition of random closed sets, and provides the natural tool to study the occlusion phenomenon, an essential ingredient in the formation of visual images. We generalize certain results of G. Matheron and, in particular, compute the probability of n compact sets being included in visible parts. This result characterizes the distribution of the boundary of the dead leaves tessellation.

Keywords

General tessellationdead leaves modeltypical cellimage modeling

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes 52A22: Random convex sets and integral geometry 68U10: Image processing
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 38 , Issue 1 , March 2006 , pp. 31 - 46
Copyright
Copyright © Applied Probability Trust 2006 

References

Alvarez, L., Gousseau, Y. and Morel, J.-M. (1999). The size of objects in natural and artificial images. Adv. Imaging Electron Phys. 111, 167–242.Google Scholar
Ambartzumian, R. V. (1974). Convex polygons and random tessellations. In Stochastic Geometry, eds Harding, E. F. and Kendall, D. G., John Wiley, New York, pp. 176–191.Google Scholar
Baccelli, F. and Bremaud, P. (2002). Elements of Queuing Theory (Appl. Math. 26), 2nd edn. Springer, Berlin.Google Scholar
Cowan, R. (1980). Properties of ergodic random mosaic processes. Math. Nachr. 97, 89–102.Google Scholar
Cowan, R. and Tsang, A. K. L. (1994). The falling-leaves mosaic and its equilibrium properties. Adv. Appl. Prob. 26, 54–62.Google Scholar
Cowan, R. and Tsang, A. K. L. (1995). Random mosaics with cells of general topology. Res. Rep. 89, University of Hong Kong.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Gille, W. (2002). The set covariance of a dead leaves model. Adv. Appl. Prob. 34, 11–20.CrossRefGoogle Scholar
Goffman, C. and Pedrick, G. (1975). A proof of the homeomorphism of Lebesgue–Stieltjes measure with Lebesgue measure. Proc. Amer. Math. Soc. 52, 196–198.Google Scholar
Gousseau, Y. (2002). Texture synthesis through level sets. In Texture 2002 (Proc. 2nd Internat. Workshop Texture Anal. Synthesis, Copenhagen, 2002), 5pp. Available at http://www.cee.hw.ac.uk/∼texture2002/index.html#ab044.Google Scholar
Gousseau, Y. and Roueff, F. (2003). The dead leaves model: general results and limits at small scales. Preprint. Available at http://arxiv.org/abs/math.PR/0312035.Google Scholar
Jeulin, D. (1989). Morphological modeling of images by sequential random functions. Advances in mathematical morphology. Signal Process. 16, 403–431.CrossRefGoogle Scholar
Jeulin, D. (1996). Dead leaves models: from space tessellation to random functions. In Proc. Internat. Symp. Adv. Theory Applications Random Sets (Fontainebleau, 1996), ed. Jeulin, D., World Scientific, River Edge, NJ, pp. 137–156.Google Scholar
Jeulin, D., Villalobos, I. T. and Dubus, A. (1995). Morphological analysis of UO2 powder using a dead leaves model. Microsc. Microanal. Microstruct. 6, 371–384.Google Scholar
Kendall, W. S. and Thonnes, E. (1999). Perfect simulation in stochastic geometry. Pattern Recognition 32, 1569–1586.CrossRefGoogle Scholar
Lee, A. B., Mumford, D. and Huang, J. (2001). Occlusion models for natural images: a statistical study of a scale-invariant dead leaves model. Internat. J. Computer Vision 41, 35–59.CrossRefGoogle Scholar
Månsson, M. and Rudemo, M. (2002). Random patterns of nonoverlapping convex grains. Adv. Appl. Prob. 34, 718–738.CrossRefGoogle Scholar
Matheron, G. (1968). Modèle séquentiel de partition aléatoire. Tech. Rep., Centre de Morphologie Mathématique, Fontainebleau.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, Chichester.Google Scholar
Mecke, J. (1980). Palm methods for stationary random mosaics. In Combinatorial Principles in Stochastic Geometry, ed. Ambartzumian, R. V., Armenian Academy of Sciences, Erevan, pp. 124–132.Google Scholar
Møller, J. (1989). Random tessellations in R d . Adv. Appl. Prob. 21, 37–73.Google Scholar
Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edn. John Wiley, Chichester.Google Scholar
Ruderman, D. L. (1997). Origins of scaling in natural images. Vision Res. 37, 3385–3398.Google Scholar
Serra, J. (1982). Image Analysis and Mathematical Morphology. Academic Press, London.Google Scholar
Stoyan, D. (1986). On generalized planar tessellations. Math. Nachr. 128, 215–219.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Weiss, V. and Zähle, M. (1988). Geometric measures for random curved mosaics of R d . Math. Nachr. 138, 313–326.Google Scholar